Dispersion cubic crystals

The type of crystalline structure that is formed depends on the concentration of the particles as well as the magnitude of the Debye-Hiickel thickness. For large Debye-Hiickel thicknesses a body-centered cubic crystal is formed, whereas for smaller values a face-centered cubic crystal is preferred. An example of the latter observed experimentally in a dispersion of latex spheres is shown in Figure 13.3. Note that this crystallization phenomenon is analogous to crystallization of simple atomic fluids, as is evident from Figure 13.3a, which shows the coexistence of a crystal with a liquidlike structure. 

FIG. 13.4 Stereo pairs of colloidal dispersions generated using computer simulations, (a) Polystyrene latex particles at a volume fraction of 0.13 with a surface potential of 50 mV. The 1 1 electrolyte concentration is 10 7 mol/cm3. The structure shown is near crystallization. (The solid-black and solid-gray particles are in the back and in the front, respectively, in the three-dimensional view.) (b) A small increase in the surface potential changes the structure to face-centered cubic crystals. (Redrawn with permission from Hunter 1989.)... 

The Dispersion of Long-Wave Polar Optical Phonons in Diatomic Cubic Crystals... 

Fig. 4. Polariton dispersion curves for a cubic crystal with three infrared active phonons 17>...

The top curve on the left-hand side of Figure 5.17 shows our qualitative dispersion curves for the dxz-yi band and dxz band from F to A to W. Note that, although there are no numerical values for the energies, the dxz band has been placed lower than the 4c2-y2 band. That is, the zero of energy for each band is not equal. A cubic crystal held would be expected to have this effect in a solid, just as in a molecule. The top curves in each diagram on the right-hand side show the corresponding calculated dispersion curves in TiO and VO. 

For pure elemental semiconductors like silicon, the strong electronic absorption at energies above Eg produces a small non-linear dispersion of the refractive index below Eg in silicon, n = 3.57 near Eg at room temperature (RT) and it steadily decreases to 3.42 for wavelengths near 12 pm and stays close to this value down to radio frequency energies (see also [20]). For these elemental crystals, the dielectric constant at energies below Eg is real and equal to n2. The refractive index is isotropic for cubic crystals, but for crystals with one anisotropic axis, like those of the wurtzite type, the refractive index for the electric field component of the radiation parallel to this axis (n//) is slightly different from that for the component perpendicular to this axis (njJ. 

Fig. 4.3. The dispersion of polariton in cubic crystals. Nongyrotropic crystals (a) The dependences of exciton and photon energy on wavevector, the retardation neglected (b) the same but with retardation taken into account. The symbols and L indicate longitudinal and transverse polarization of excitons (c) retardation neglected but dependence of the exciton energy on the wavevector taken into account here and in (d), (e), and (f) only the lower branch of the polaritons shown (d) the retardation and dependence of exciton energy on wavevector are taken into account. Gyrotropic crystals (e) Dispersion of excitons in the cubic gyrotropic crystals if retardation is neglected (f) the same when retardation is also taken into account Aq denotes the position of the bottom of the polariton energy.

However, this formula, which expresses the dielectric constant of the medium in terms of the polarizability of an individual molecule, is only a rough approximation even for cubic crystals with the van der Waals forces acting between the isotropic molecules. For instance, the formula takes no account of spatial dispersion. Moreover, it does not take into account the contribution of the higher multipoles to the energy of the intermolecular interaction which is important at distances of the order of lattice constant. 

As was shown by Born and Huang (4) in cubic crystals with one molecule per unit cell the tensor Q"f is reduced to the scalar QtJ = (An/ yv)dlJ where v is the volume of the unit cell, if we ignore spatial dispersion. Moreover, a - = aSij so that eqn (5.3) yields the tensor Ai3 = Adij where A = [1 — (47ra/3v). Substituting this expression into eqn (5.6) we obtain c,3 = eSl3 where... 

The dielectric tensor in a cubic crystal is reduced, as is well known, to the scalar dielectric function e(u>) when spatial dispersion is neglected. In the region of the band of two-particle states, this function can be presented in the form... 

Polaritons in cubic crystals can be transverse or longitudinal and as we neglected the spatial dispersion, the polariton dispersion law, i.e. the dependence... 

Let us assume that the crystal surface coincides with the plane z = 0, that the half-space z > 0 is vacuum (e j = Sij), and that the half-space z < 0 is a cubic crystal with a dielectric function e(w) (spatial dispersion is neglected so far). Hence, we are interested in the solutions of Maxwell s equations that vanish as z —> oo we shall look for them in the following form ... 

Dispersion curves of longitudinal and transverse osciQations along direction a type of that is L[lll] and T[lll] for potassium are presented in Figure 12.4. This direction is chosen because the distance between neighboring atoms is minimal along it in the body-centered cubic crystal lattice. Similar curves are typical for other alkali elements. 

True dispersion forces are present to all appearances in the lattices of inert gases. These substances crystallize in very highly symmetrical arrangement—cubic face-centered— both the lattice constants and the heats of sublimation are easily obtainable experimentally. Eisenschitz and London, by employing the calculation mentioned on page 93 of the quantum mechanical component of the van der Waals forces for the lattice energy of a cubic crystal, which coheres only through such force effects, have derived an equation of the form... 

Fig. 2.10 Schematic representation of the bulk (shaded area) and sur-foce (dashed line) phonon dispersions for a (110) surface of a cubic crystal. The symmetry lines for the first 2D Brillouin zone are shown in Fig. 2.7a.

Besides the discrepancies between calculated and observed dispersion curves as illustrated in Fig.4.5, there are other deficiencies of the rigid-ion model. Since the model is based on central forces, it predicts the Cauchy relations which for a cubic crystal are C 2 = 44- glance at Table 3.2 shows that these relations are only approximately satisfied for the alkali halides. 

Surface modes can be clearly identified in the dispersion relations ft>(qn) when they appear in regions where no bulk bands appear. Similar to the identification of surface electronic states, the projected bulk modes form the bulk phonon bands in the surface Brillouin zone, as shown in Figure 9.46. In the bulk case, there are three acoustic phonon bands and 3(S-1) optical phonon bands, with S as the number of atoms in the primitive unit cell of the bulk crystal. Along high-symmetry directions in the bulk, such as the (100) or (111) directions in cubic crystals, the phonons can be classified either as transverse or longitudinal, depending on whether or not their displacements are perpendicular or parallel to the direction of the 3D wave vector. 

In the reaction flask, lithium aluminum hydride (14.4 g, 38 mmol) and benzyltriethylammonium chloride (2.9 g, 13 mmol) are dispersed in tetralin (450 mL) with stirring imder nitrogen. The product of Step 4 (63 mmol of (BrSiH2)4C and toluene) is added dropwise with stirring. The mixture is stirred for 70 h at room temperature, followed by 5 h at 60°C. Toluene and tetra(silyl)methane are then condensed out of the reaction mixture kept at 60°C in a vacuum and collected in a trap held at 10°C. (Small amounts of silane are trapped in a small final trap at -196°C. From there it is allowed to escape through a safety valve to burn upon careful warming of the trap.) The liquid is subjected to fractional distillation and the product collected at 89.5°C, 4.3 g yield (50%). It solidifies in the receiver as colorless cubic crystals, mp 38°C. NMR (benzene-de) 3.84 (s, SiHs) -39.0 (t dez), J(SiC) 31.3, J(CSiH) 5.5) Si -47.8 (q dez), J(SiH) 205.7, J(SiCSiH) 4.6). MS 137-124 (CSi4HJ. 

---